Page 224 - DMTH503_TOPOLOGY

Topology

Notes 
Then given> 0, there exist a positive integer n such that m, nn d(x , x) < and d(x , x)
o o m 2 n
  
< . Therefore, m, nn d(x , x )d(x , x) + d(x, x ) < + < =.
2 o m n m n 2 2
Hence, <x > is a Cauchy sequence.
n

Note The converse of this theorem is not true i.e., Cauchy sequence need not be
convergent.

To prove this, consider the following example.
Let X =  – {0}.
Let d(x, y) = |x – y|

1
Consider the sequence x = , n
n
n
We shall show that
<x > is a Cauchy sequence but it does not converge in X. Let> 0 be given and n be a positive
n o
2
integer such that n > .
o

Now d(x , x ) = x - x
m n m n
= x + ( x )
-
m n
= x + x
m n
1 1
= +
m n
2 1 
If mn m > and so <
o
 m 2
1 
Similarly, <
n 2

1 1  
 d(x , x ) + < + = 
m n
m n 2 2
Thus d(x , x ) < .
m n
Hence <x > is a Cauchy sequence.
n
Clearly, the limit of this sequence is 0 (zero) which does not belong to X.

Thus x does not converge in X.
n
Example 1: Let <a > be a Cauchy sequence in a metric space (X,) and let <b > be any
n n
1
sequence in X s.t. (a , b ) <  nN.
n n
n

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